Expected Value: Dice

Calculate expected value for dice scenarios

Calculator Dice Coins Marbles Cards Learn

Single Die1 Die Sum of Two DiceSum Max of Two DiceMax

Expected Maximum of Two Dice

and

Question

When you roll two dice and take the maximum, what is the expected value?

Example: If you roll 2 and 5, max(2, 5) = 5.

Step 1: Find P(Max = k)

For the maximum to equal k, we need at least one die to show k, and neither die to show more than k.

Key insight:

Since dice are independent, we can multiply probabilities.

Step 2: Build the Distribution

Max = k	2k - 1	P(Max = k)	k × P(k)
1	1
2	3
3	5
4	7
5	9
6	11

Check: 1 + 3 + 5 + 7 + 9 + 11 = 36 (probabilities sum to 1).

Step 3: Calculate E(Max)

Answer

E(Max) =

On average, the maximum of two dice is about 4.47.

Comparison: Max vs Sum vs Single

E(Single Die)

3.5

E(Max of Two)

4.47

E(Sum of Two)

The expected maximum is between the single die (3.5) and the sum divided by 2 (3.5). Taking the max gives you an advantage of about 1 point on average!

Why This Matters

In games with "advantage" mechanics (roll twice, take higher):

Your expected roll increases from 3.5 to 4.47
That is a 28% improvement!
Higher values become much more likely (P(6) = 11/36 = 30.6% vs 16.7%)